Linear systems are a class of systems
rather than having a specific inputoutput relation. Linear
systems form the foundation of system theory, and are the most
important class of systems in communications. They have the
property that when the input is expressed as a weighted sum of
component signals, the output equals the same weighted sum of
the outputs produced by each component. When
S·
S
·
is linear,
S
G
1
x
1
t+
G
2
x
2
t=
G
1
S
x
1
t+
G
2
S
x
2
t
S
G
1
x
1
t
G
2
x
2
t
G
1
S
x
1
t
G
2
S
x
2
t
(5)
for all choices of signals and gains.
This general inputoutput relation property can be manipulated
to indicate specific properties shared by all linear systems.

SGxt=GSxt
S
G
x
t
G
S
x
t
The colloquialism summarizing this property is "Double the
input, you double the output." Note that this property is
consistent with alternate ways of expressing gain changes:
Since
2xt
2
x
t
also equals
xt+xt
x
t
x
t
,
the linear system definition provides the same output no
matter which of these is used to express a given signal.

S0=0
S
0
0
If the input is identically zero for all
time, the output of a linear system must be
zero. This property follows from the simple derivation
S0=Sxt−xt=Sxt−Sxt=0
S
0
S
x
t
x
t
S
x
t
S
x
t
0
.
Just why linear systems are so important is related not only
to their properties, which are divulged throughout this
course, but also because they lend themselves to relatively
simple mathematical analysis. Said another way, "They're
the only systems we thoroughly understand!"
We can find the output of any linear system to a complicated
input by decomposing the input into simple signals. The
equation above
says that when a system is linear, its output to a decomposed
input is the sum of outputs to each input. For example, if
xt=e−t+sin2π
f
0
t
x
t
t
2
f
0
t
the output
Sxt
S
x
t
of any linear system equals
yt=Se−t+Ssin2π
f
0
t
y
t
S
t
S
2
f
0
t
"Electrical Engineering Digital Processing Systems in Braille."